Interactions of pulsatile upstream forcing with flow-induced oscillations of a collapsed tube: mode-locking

Interest in flow through collapsible tubes derives from their prevalence in the human body. In the circulation at least, the driving pressure for flow is pulsatile, and it is accordingly appropriate to investigate the response of a collapsible tube to a pulsatile head. A servo-controlled hydraulically powered piston pump was used to add pulsation to the head propelling aqueous flow through a silicone rubber tube. The pump was commanded to execute a sinusoidal variation of piston position at various frequencies on either side of the natural frequency of an operating point where slow (3.75 Hz) repetitive self-excited oscillations in the absence of such forcing could be guaranteed. The resulting time series were characterised by the ratio of the number of collapse cycles to the number of forcing cycles in one overall interaction waveform period. Nonlinear interaction rather than linear superposition was dominant. Many different mode-locked interactions were recorded, and are here systematised in Farey series and plots which show the extent of entrainment. A few of the time series were aperiodic over the time scales investigated, suggesting possible chaos.